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### Type ${\text{III}}_{0}$ cocycles without unbounded gaps

Commentationes Mathematicae Universitatis Carolinae

An example of type III${}_{0}$ cocycle without unbounded gaps of an ergodic probability measure preserving transformation will be shown.

### On measure theoretical analogues of the Takesaki structure theorem for type III factors

Colloquium Mathematicae

The orbit equivalence of type $II{I}_{0}$ ergodic equivalence relations is considered. We show that it is equivalent to the outer conjugacy problem for the natural trace-scaling action of a countable dense ℝ-subgroup by automorphisms of the Radon-Nikodym skew product extensions of these relations. A similar result holds for the weak equivalence of arbitrary type $II{I}_{0}$ cocycles with values in Abelian groups.

### Mod 2 normal numbers and skew products

Studia Mathematica

Let E be an interval in the unit interval [0,1). For each x ∈ [0,1) define dₙ(x) ∈ 0,1 by $dₙ\left(x\right):={\sum }_{i=1}^{n}{1}_{E}\left({2}^{i-1}x\right)\left(mod2\right)$, where t is the fractional part of t. Then x is called a normal number mod 2 with respect to E if ${N}^{-1}{\sum }_{n=1}^{N}dₙ\left(x\right)$ converges to 1/2. It is shown that for any interval E ≠(1/6, 5/6) a.e. x is a normal number mod 2 with respect to E. For E = (1/6, 5/6) it is proved that ${N}^{-1}{\sum }_{n=1}^{N}dₙ\left(x\right)$ converges a.e. and the limit equals 1/3 or 2/3 depending on x.

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